Let $\mathcal L, \mathcal L^*: \Theta \times \mathcal A \to \mathbb R$ be functions. When can $\mathcal L$ be expressed as the convolution of $\mathcal L^*$ with some third function $U$? That is, when does the following hold:
\begin{align} \mathcal L (\theta, a) &= (\mathcal L^* \ast U) (\theta, a) \end{align}
In practice $\mathcal L$ and $\mathcal L^*$ will be nice (often convex). If regularity conditions on $\mathcal L, \mathcal L^*$ are necessary for this to hold, I'm curious what they are.
Second question: Given $\mathcal L$ and $\mathcal L^*$, what can we say about $U$?
For example, let $\mathcal L$ be the weighted log loss, and let $\mathcal L^*$ be the log loss. Given the exact form of $\mathcal L$ and $\mathcal L^*$ can we recover the weights in the weighted log loss? In general, when can a loss function be represented as the convolution of a utility function with the log loss?
So far I have come across the convolution theorem and Representations of functions by convolutions by Rudin. The convolution theorem seems to imply that if $\mathcal L$ can be expressed as $\mathcal L^* \ast U$ (again recalling that both $\mathcal L$ and $\mathcal L^*$ are fixed), then we can use Fourier transforms and inverse Fourier transforms to recover $U$ (although the precise conditions are a bit hazy to me). Rudin's paper seems to imply that $\mathcal L$ can be written as the convolution of two $L_1$ functions, but that $\mathcal L^*$ is not necessarily one of these functions.
Here's what I can piece together so far. For given $\mathcal L$ and $\mathcal L^*$, note that
\begin{align} \mathcal L = \delta \ast \mathcal L = \mathcal L^* \ast \mathcal {L^*}^{-1} \ast \mathcal L \end{align}
where $\ast$ indicates convolution, $\delta$ is the Dirac delta function, and $\mathcal {L^*}^{-1}$ is the multiplicate inverse of $\mathcal L^*$ in the convolution measure algebra. Then we take $U = \mathcal {L^*}^{-1} \ast \mathcal L$.
Now we want to know when $\mathcal {L^*}^{-1}$ exists (and also when it's unique). This answer provides a condition for the existence of $\mathcal {L^*}^{-1}$ when $\mathcal L^*$ is a compactly supported Schwartz distribution. This is all way nastier than I anticipated.
My guess is that most loss functions would satisfy these conditions if they were truncated to compact support, but ugh checking the conditions seems unpleasant.